In classical data analysis, variables describing entities are usually mono-valued, that is, each entity takes exactly one value for each variable at each point in time. However, this model is too restrictive to represent more complex data where variability and/or uncertainty might be inherent to each observation. This is the case of interval data where the observed values of the variables are intervals of IR. Interval-valued data may be represented by the lower and upper bounds of each observed interval or, equivalently, by its center and radius. When the interval-valued symbolic data are collected as an ordered sequence through time or any other dimension, they form an interval time series (ITS). Time series models such as ARIMA processes are appropriate for single-valued data sets and therefore we propose a new approach for ITS, namely using Space-time autoregressive models (STAR) for such data, thus taking into account the existence of contemporaneous correlation or dependence between the intervals' lower and upper bounds (or center and radius).
We start by setting up the bivariate STAR model for the ITS bounds and derive the corresponding bivariate model for its center and radius which is a Structural Vector Auto-regressive model (SVAR) with the same order. The parameters of the latter are functions of those in the former. Important particular cases and their consequences are also analyzed. Prediction of the ITS bounds from the respective STAR model and of the center and radius from the respective SVAR model is also discussed. An application of this approach is given where the STAR model for the interval bounds is estimated and checked for adequacy. The corresponding model for the center and radius is derived and its parameter estimates are computed from those of the STAR model. Next, the ITS values were forecast for several periods (out-of-sample forecasts) showing good predictive performance. Finally, the equivalence between the forecasts obtained from the ITS bounds and the ITS center and radius is shown.
Arroyo, J., González-Rivera, G., Maté, C. (2010). Forecasting with interval and histogram data. Some Financial applications. In: Handbook of Empirical Economics and Finance, Ullah, A. et al (Eds.), Chapman and Hall/CRC, New York.
Billard, L., Diday, E. (2003). From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis, Journal of the American Statistical Association 98, 470-487.
Cressie, N.A.C. (1993). Statistics for Spatial Data, Wiley, New York.
Diday, E. and Noirhomme, M. (2008). Symbolic Data and the SODAS Software, Wiley, Chichester.
Finkenstädt, B., Held, L. and Isham, V. (Eds.) (2007). Statistical Methods for Spatio-Temporal Systems, Chapman and Hall/CRC, London.
Teles, P. and Brito, P. (2005). Modelling interval time series data, Proc. 3rd IASC World Conference on Computational Statistics and Data Analysis, Cyprus.
Keywords: Interval bounds, center and radius; Interval time series; Prediction; Space-time AR model
Biography: Paulo Teles has a Ph.D. in Statistics from Temple University (Philadelphia, USA) and is currently Assistant Professor in the School of Economics of the University of Porto (Portugal). His research interests are Time Series Analysis, Forecasting and Time Series Econometrics. He was Director of the Northern Regional Office of the Portuguese National Statistical Institute in 2001-2004 and their consultant statistician in 2005-2010. He has been in the organizing committees of different conferences, namely the International Symposium on Forecasting in 2000 (where he was the Local Chair) and the Compstat 2008. He is a member of the of Artificial Intelligence and Decision Support Research Group of the University of Porto (Portugal), a member of the American Statistical Association and of the Portuguese Statistical Society.